Question: The day length in Juneau, Alaska, varies over time in a periodic way that can be modeled by a trigonometric function. Assume the length of the year (which is the period of change) is exactly $365$ days. The longest day of the year is June $21$, and it's $1096.5$ minutes long. The shortest day of the year is half a year later, and it's $382.5$ minutes long. Note that June $21$ is $171$ days after January $1$. Find the formula of the trigonometric function that models the length $L$ of the day $t$ days after January $1$. Define the function using radians. $ L(t) = $
Answer: Let's start by writing a formula for the length of the day $u$ days after June $21$. Both sine and cosine can be used to model periodic contexts. We can decide which is better fitting by considering the $y$ -intercept. The sine function intercepts the $y$ -axis at its midline, and the cosine function intercepts the $y$ -axis at its peak. Day length is at its peak at time $u = 0$, so we'll use a cosine function to model the length of the days (because cosine functions also have a peak at $u= 0$ ). The day length in Juneau has period $365$ days. Its midline is halfway from its maximum value to its minimum value, or $\dfrac{382.5 + 1096.5}{2} = 739.5$ Its amplitude is half the difference between its maximum and minimum values, or $\dfrac{1096.5 - 382.5}{2} = 357$ Since the ordinary cosine function $f(u) = \cos u$ has period $2\pi$, midline $y = 0$, and amplitude $1$, we can stretch it horizontally by a factor of ${\dfrac{365}{2\pi}}$, stretch it vertically by a factor of ${357}$, and move it up ${739.5}$ units: $ L(u) = {357}\cos\left({\dfrac{2\pi}{365}}u\right) + {739.5}$ Since June $21$ is $171$ days after January $1$, the day that is $t$ days after January $1$ is $t - 171$ days after June $21$, so $u = t - 171$ : $ L(t) = {357}\cos\left({\dfrac{2\pi}{365}}(t - 171)\right) + {739.5}$ The function $ L(t) = {357}\cos\left({\dfrac{2\pi}{365}}(t - 171)\right) + {739.5}$ has period $365$, amplitude $357$, and midline $y = 739.5$, and it reaches its maximum at time $t = 171$, so it's a good model of the length of the day in Juneau.